one-field

To be is, minimally, to be self-present. Anything that exists already affects itself by being there. Interference is the physical image of this. What is said to exist is what carries non-zero interference weight. The framework observes what emerges from interference.

A substrate self-interferes. Exactly one family of standing-wave patterns closes to give our universe: the Standard Model, gravity, and the dark sector.

Concretely, from one dimensional anchor (the electron mass) and the closure demand whose unique solution is the algebra, the code produces the other eight fermion masses, the CKM and PMNS scorecards, the electroweak scale, the fine-structure and strong couplings, Newton's constant, the baryon asymmetry, and the dark-matter fraction, with zero tunable dimensionless parameters. Each result carries an explicit epistemic tag: theorem, enumeration, identification hypothesis, coordinate convention, arithmetic observation, or cited import with a stated band. The runner prints the scorecard and derivation graph. The substrate simulations are the physical-realization program: they test emergence and supply no scorecard input.

Run it

git clone https://github.com/kuwrom/one-field.git
cd one-field
pip install -r requirements.txt
python3 interference/run.py   # full derivation, prints scorecard and derivation graph
pytest -q                     # 170 tests (155 fast + 15 slow behind -m slow)

Python 3.10+, NumPy, SciPy. Twenty seconds to a minute on one core.

Masses (one anchor in, eight out, max error 0.4%)

Particle	Predicted	PDG	Error	Method	Status
e	0.5109990 MeV	0.5109990 MeV	anchor	CODATA, sole dimensional input	input
μ	105.6584 MeV	105.6584 MeV	+0.0000%	Brannen Z₃ (predicted)	Thm + IH
τ	1776.909 MeV	1776.930 MeV	−0.0012%	Brannen Z₃ (predicted)	Thm + IH
u	2.158 MeV	2.16 MeV	−0.1%	(38/9) m_e	Thm + IH + conv
d	4.713 MeV	4.70 MeV	+0.3%	(83/9) m_e	Thm + IH + conv
s	93.8 MeV	93.5 MeV	+0.4%	Q₀+h₁₀/K³ + bridge	Thm + IH + conv
c	1273.8 MeV	1273 MeV	+0.06%	(217/18) m_μ	Thm + IH + conv
b	4193.8 MeV	4183 MeV	+0.3%	Q = Q₀ + h₁₁/K³	Thm + IH + conv
t	172.51 GeV	172.57 GeV	−0.036%	(1165/12) m_τ	Thm + IH

Fine print: mass coordinates and status labels

Status labels. Thm = executable theorem in this repo. IH = identification hypothesis (a mathematical object matched with a physical one, separately falsifiable). conv = stated scheme convention. Full taxonomy in What is derived vs identified.

The framework produces one completed reading per fermion, every echo included. It does not run to a scale, so each comparison names the coordinate attached to the external reference. Unconfined fermions (leptons, top) sit at the propagator pole. Confined heavy quarks (c, b) sit at the self-scale m(m). Light quarks (u, d, s) sit at the PDG MS-bar(2 GeV) coordinate. The alternative charm reference near 1.67 GeV moves the charm Koide residual from +0.06% to roughly −24% without any change to the framework, and the Koide relation among the measured lepton masses, transported to M_Z, holds only at the 0.17% level. The claim is always the completed reading at its stated coordinate, never a relation at every scale. The invariant content of the light sector is its RG-invariant ratios, which carry no coordinate at all: m_u/m_d = 38/83 (−0.9σ vs PDG 0.473(17)), m_s/m_ud = 27.318 (+0.2σ vs 27.30(8)), Q_ellipse = 22.383 (+0.4σ dispersive, −1.7σ lattice, and the two references disagree). See masses.py for the full treatment.

Lepton mass chain. The calculation uses Brannen's 2006 circulant form. The G₂ → SU(3) Fano-plane calculation gives the Clebsch-Gordan ratio √2. The closing lane's SU(3)₃ modular data have canonical positive spherical dimension d(1,0) = 2. The channel-reading map combines these data with two cycle orientations and gives B/A = √2. The circulant identity then gives Q₀ = 2/3. The lightest triality-changing primary has Sugawara weight h = 2/9, and the mass-phase map sets θ = h. This is the sole positive, nondegenerate value in the committed angle menu. The reading is one ledger, every interference layer included: the forced depth-4 EM backreaction, flavour-dependent so it survives in the mass ratios, contributes −9.8 ppm on the μ/e ratio, and the measurement resolves it (masses.py). The framework computes the physical particle with its electromagnetic self-interference included, which is why the propagator-pole entry is the comparison coordinate. The Koide relation and Brannen parameterization predate this implementation, so the lepton scorecard is postdictive. With m_e as the dimensional anchor, the calculation reproduces m_μ to 0.0000003% and m_τ to 0.0012%.

What mass is

A particle is a self-trapped patch of the substrate, a lump whose internal phase circulates at one fixed frequency. That rate is the mass, E = ħω. A heavier particle is a faster internal clock, and nothing else about it is "heavy".

The mass construction has three clock settings because it uses three child labels with Z₃ cyclic symmetry (a → b → c → a). A graph can of course also have self-loops or two-cycles (the Web uses both), so three is a framework commitment, not a graph-theoretic minimum. When the bias that singles out one child label is erased (protected forgetting), the resulting Z₃-circulant has three eigenstates identified with the electron, muon, and tau generations. The circulant fixes the mass ratios. The electron fixes their dimensional unit. m_μ and m_τ are then determined by m_e alone.

Any configuration that closes to give the leptons automatically produces quarks and gravity from the same structure. The universe does not have to get lucky twice. The lepton standing wave closes without engaging the embedding and sits at the confinement scale. The quark standing wave closes through the embedding and sits at the electroweak scale. Three generations of each are the same Z₃, not triality. The mass sector is one generator plus a grammar: the lepton circulant is the generator, and the quark masses are arithmetic on it. The full mass chain is in interference/__init__.py.

Why octonions

The searchlight is organized around the three-label closure. Its starting premise is that a closed record is a finite-dimensional unital real composition algebra: records compose bilinearly and their positive-definite norm, read as probability weight, is multiplicative. Hurwitz's classification then leaves exactly R, C, H, and O. Relative to the real-amplitude baseline, the three non-real stages C, H, and O give searchlight depths 1–3. The next Cayley–Dickson algebra fails the criterion through zero divisors. This is a complete derivation from the framework's ledger premise, not a claim that Hurwitz, in isolation, is a theorem about physical self-reference. searchlight/octonions.py verifies bilinearity, norm multiplicativity, the explicit sedenion zero divisor, and the Fano/MTC channel map. The canonical Web then carries an explicit depth-3 e↔q loop ledger, and every depth-4 backreaction names and actually reads that parent.

Mixing, couplings, gravity, baryogenesis, dark sector

Sector	Result	Reference	Status
Electroweak scale v_EW	246.219645 GeV	246.219651 GeV (from G_F)	−0.02 ppm
Fine structure 1/α(0)	137.035999050	Berkeley Cs 137.035999046(27)	+0.13σ vs Cs, −14σ vs Rb 2020. The commitment is to the Cs side of the 5.5σ dispute. The dispute resolving toward Rb kills it
Higgs transport	lambda(m_t) = 0.12651	Internal transport	Candidate bridge boundary in the imported beta coordinate. The depth-4 Web reading is unassigned
CKM (15 comparison coordinates, correlations unmodeled)	diagonal pull score/coordinate = 0.21	PDG 2024	max marginal pull 0.7σ. The reference block mixes correlated global-fit outputs with a separately quoted phase constraint. Without a joint covariance model this is a descriptive score, not χ². Where mixing lives is itself an executable theorem: two Z₃-equivariant transports commute, giving V = 1 exactly, so mixing is not a sector transport. It enters as four readings of the solved D⁽⁶⁾* boundary, one per face, rigid engine constants with no continuous freedom (mixing_z3_dichotomy.py). Each reading carries its own grade, from computed (η̄ = π/9, the braiding monodromy) through enumerated (λ, the unique 1σ hit over the engine's echo menu, frozen in CI) to identified (the attachment to the physical Wolfenstein coordinates). The per-face grading is in mixing.py
PMNS (3 correlated angles)	diagonal pull score/angle = 0.00	Frozen NuFit 6.0 NO snapshot, IC19 without SK atmospheric data	Identified conjugation-orbit TBM map with U_e = R₂₃(−φ)R₁₃(θ_C,δ). The construction consumes no new engine numbers: the TBM block is computed from the conjugation-orbit-compressed fusion matrices, and the corrections are functions of the derived mass spectrum, φ = √(m_e/m_μ)/√d₁₀, and the echoed λ. Symmetric marginal errors and no covariance make this descriptive rather than χ². The current −arg(U_e3) = 77.0° is convention-dependent and is not reported as the physical δ_CP
Strong coupling α_s(M_Z)	0.1184	PDG 0.1180(9)	+0.4σ (π/32 exact at μ* = 253.5 GeV)
Weinberg angle sin²θ_W	0.231285	PDG 0.23129(4)	pull −0.11σ
W mass M_W	80.356 GeV	80.3692(133) world avg	pull −1.00σ
Z mass M_Z	91.189 GeV	91.188 GeV	+0.0010%
Newton's constant G_ind/G_N	0.999999917 (apportioned)	1	Internal closure under the Planck/healing-length identification: Σ_target = 2πd₁₁ = 6π. The probe's two factorizations are the same cutoff identity. Phase readings: 1.0000 (UV), 1.0134 (broken)
Baryon asymmetry η_B	6.178e-10	Planck 6.12e-10	+0.9%
Dark / baryon ratio Ω_DM/Ω_b	2π − 1 = 5.2832	Planck 5.364	pull −1.26σ

Joint cosmology closure. The two cosmology channels are derived independently (η_B from the G₂ instanton, Ω_DM/Ω_b from bridge venting) but they must agree jointly with the absolute dark-matter density. They do. η_B = 6.178e-10 → Ω_b h² = 0.02256 (+1.2σ vs Planck), and Ω_DM h² = (2π−1) · Ω_b h² = 0.1192 (−0.7σ vs Planck 0.1200(12)). Two independent channels, one consistent cosmology. The BBN conversion η₁₀ = 273.9 Ω_b h² is cited, not derived: it is a change of coordinate on a printed number, and no engine step consumes it.

Conditional structural constraint and cosmology bookkeeping. If the two F₄ singlets are right-handed neutrinos and the mass mechanism is a minimal type-I seesaw with no other light-neutrino mass operators, then the light mass matrix has rank at most two. Under that assumption at least one state is massless. The ordering remains unselected (m₁ = 0 for normal or m₃ = 0 for inverted). Separately, (3/8π) M_Pl² H₀² is the Friedmann critical density, not ρ_Λ. The current dark-energy density is Ω_Λ times it. Jacobson's thermodynamic derivation leaves Λ as an integration constant, and CKN saturation is an identification hypothesis.

What gravity is

Mass, above, was a clock. Gravity is the price of the ticking, seen from outside. In a standing wave the substrate circulates in place. That circulation takes up room in the shared background and depletes it, and the knot and its depletion form together (tests/probes/skyrmion_3d.py). The soliton casts a shadow. A depletion is a deficit, and a deficit is a pull. Every standing wave pushes on the same shared background, and their depletions compose: a planet's worth of circulations makes one giant shadow, a unified ventilation that all the particles in the planet breathe through. Having mass and sourcing gravity are one event.

Gravity is not a fifth force layered on top of matter. It is the shape matter carves into the background while ventilating itself, and the bias other matter rides into. The full treatment is in gravity.py and interference/__init__.py.

The normalisation is audited from both directions. The minimal matter spectrum alone gives the wrong sign, Σ_min ≈ −10.2, repulsive gravity. The 182 bridge channels, a count forced by the wiring winner and entering linearly with no adjustable multiplicity, supply the +29.126 that closes G_ind/G_N to 8×10⁻⁸. The unfactored cutoff schemes miss on their own: t₀ = ξ₀² gives G_ind/G_N = 1/12, 2ξ₀² gives 1/6, ℓ_Pl² gives 1/3, so the target Σ = 2πd₁₁ sits on no unfactored scheme. The closure is internal, and its one measured contact is the derived M_Pl at +22 ppm (sakharov_normalisation.py).

What closes

A substrate self-interferes, and many patterns form. Most do not close: lanes that make electron-like things whose ratios fail to continue, quark-like things that never bind. Six consistency gates eliminate them. Only one lane closes all six: protected G₂ forgetting applied exactly once. The gates and their scan are the searchlight's bookkeeping, not a mechanism. The substrate closes or fails to close, and the scan is the computed record that nothing else passes, run on Lie-table invariants alone with its one genuine contender, E₈ ⊃ E₇ × A₁, printed beside the winner (embedding_uniqueness.py). The seven-branch gauntlet is the computed record of the shorter trees. The 504 hits reported later are mass grammar realizations within this fixed lane, not additional closure lanes.

That surviving lane sits inside one algebra:

14  +  52  +  182  =  248  =  dim(E₈)          E₈(1) ⊃ G₂(1) × F₄(1)
gauge   matter   bridge                         c: 14/5 + 26/5 = 8, exact
        + Higgs  → gravity                      248 → (14,1) ⊕ (1,52) ⊕ (7,26)

Every degree of freedom of E₈ has a physical job. Nothing is left over. The audit is in interference/__init__.py.

Quantum numbers

The E₈ → G₂ × F₄ representation theory derives the Standard Model quantum numbers directly (root.py):

Q_u  = d₁₀/d₁₁ = 2/3         (up-type electric charge)
Q_d  = 1/d₁₁   = 1/3         (down-type electric charge)
N_c  = d₁₁     = 3           (colour multiplicity)
sin²θ_W = d₁₁/(d₁₀²+d₁₁²) = 3/13   (Weinberg angle, at the embedding scale)

The charge trace Σ Q² N_c = (d₁₀²+1+d₁₁)/d₁₁ = 8/3 gives the Singh ratio (8/3)·C₂(26) = 16 that converts the G₂ coupling to the electromagnetic one: α_EM = α_G₂/16 = π/512.

CP violation traces to the oriented Fano plane: seven directed lines select one orientation sign. Reversing it flips η̄, J_CKM, J_lep, and η_B together while leaving CP-even quantities unchanged (tests/probes/orientation_bit.py). One global CP-orientation bit, not several independently inserted signs.

Strong CP: the index-one G₂ → SU(3)_c embedding preserves winding. In the no-Lagrangian ontology, the physical angle is the complete matter-coupled character θ̄, and the full record rule makes that character act on the vacuum record algebra A = 1 + B. Multiplication leaves only the trivial phase automorphism, so θ̄ = 0 mod 2π. This conclusion follows within that ontology rather than from local MTC data alone. A measured neutron EDM attributable to θ̄ ≠ 0 kills it (derivation, supporting probe).

Derivation flow

The scorecard is an executable dependency pipeline, run in order by interference/run.py:

1. wiring.py scans the finite candidate menus (lattices, conformal pairs, levels) and finds the unique closure. The depth-3 searchlight reads d₁₀ = 2, d₁₁ = 3, n₇ = 7, n₂₆ = 26, and α_G₂(M_Pl) = 1/(24π) off the winner. root.py consumes that vocabulary, and run.py asserts that both stages share the same wiring record.
2. root.py builds the recursive web, its couplings and scales, and derives M_Pl by inverting the electron anchor.
3. words.py derives the generation walk counts. masses.py derives the nine fermion masses and v_EW.
4. mixing.py computes the WZW data, exact D⁽⁶⁾* connection, and mixing scorecards. couplings.py computes α_s and α(0).
5. gravity.py computes the induced-gravity, baryogenesis, and neutrino outputs. higgs_ew.py computes the Higgs quartic boundary and electroweak observables (SM RGE transport, M_W, M_Z). dark_sector.py computes the cosmological ratios.

The substrate is the physical realization of this chain. Its probes test emergence. They do not supply scorecard inputs.

Architecture

The derivation lives in the interference/ package as one dependency chain. Package-relative imports produce one canonical module graph and one Web singleton. The scrutiny lives in tests/: freezes pin the numbers, probes cover mechanisms and falsifiers, and the substrate and audit instruments provide numerical checks. Each derivation module exposes one derive(...) → dict, and run.py calls them in dependency order. The full narrative follows that order in interference/__init__.py.

Module	Role
run.py	End-to-end runner: python3 interference/run.py
root.py	Embedding, Casimirs, central charges, the recursive echo web, M_Pl from the m_e anchor. The four integers are the depth-3 searchlight's vocabulary, read from wiring.py
masses.py	All nine masses: Brannen Z₃ leptons, F₄ quarks, v_EW
mixing.py	WZW/D⁽⁶⁾ boundary data, braiding phase, CKM parameter map, PMNS conjugation-orbit TBM construction
couplings.py	α_s (WZW cancellation at μ*), α(0) from bridge self-interference
gravity.py	Induced-gravity ledger, target factorizations, η_B, neutrinos
higgs_ew.py	SM RGE transport, Higgs quartic boundary, M_W, M_Z, import audit
dark_sector.py	Ω_DM/Ω_b = 2π − 1 from bridge venting
words.py	Generation word lemma: nimrep walk counts → quark mass bases
wiring.py	Closure-residual scan over the candidate menus. The four integers are never an input
reference.py	Comparison data, single home, quarantined by tests/test_blindness.py
searchlight/	The octonions: G₂/SU(3) CG verification, embedding uniqueness through all six gates, protected forgetting. The one subdirectory, and the engine never imports it
tests/substrate/	Z₃-NLS substrate: BdG, stability, Madelung sourcing, and the same equation in 3D
tests/audit/	The committed grammar, and the grammar run as a generator: blind gates first, data contact once

Physics FAQ

The derivation probes in tests/probes/ answer standard physics objections. Each states its own verdict and kill condition. The table below is the discoverability map.

Question	Answer	Code	Status
Why 3+1 dimensions?	The Z₃ relative sector is C², so the order parameter space is S³. π₃(S³) = Z first permits winding in 3 spatial dimensions. Topology-changing zeros of a four-real-component field have codimension 4, making them point-events exactly in D = 4	dimensionality.py	Derived
Why are solitons stable?	The bare NLS unwinds (intentional negative result in knot_charge.py). The Faddeev-Skyrme quartic stabilising term is then constructively derived from the framework's own energy functional	effective_action.py	Derived
Is there confinement?	Vortex proliferation is measured, the area-law fit beats the perimeter-law fit decisively, and the Gaussian excitation spectrum is gapped	confined_phase.py	Classical confinement signature and Gaussian gap computed
How does gravity enter?	The exact Z₃ split isolates the common mode. Its sourced response and long-wavelength dispersion are computed. Given the declared common-mode/acoustic-geometry and Planck/healing-length identifications, the standard acoustic-metric and DeWitt–Seeley heat-kernel theorems produce the induced Einstein–Hilbert term, whose coefficient is evaluated by the venting ledger	gravity.py, sakharov_normalisation.py	Complete derivation within the stated identifications
What about chirality/CP?	Closure leaves the mass-protected chiral branch, dimensionality gives 3+1D, and the embedding branch supplies the gauge representations. The radial Skyrmion–Dirac operator is built on the repository's actual π₃ hedgehog and exhibits the single localized crossing associated with unit winding. Its scanned coupling is a spectral-family coordinate, not a fitted or additional framework parameter. Independently, reversing the Fano orientation flips every computed CP-odd sign and preserves CP-even magnitudes. These branches compose under the stated fermion/knot and representation identifications	zero_mode.py, orientation_bit.py, dimensionality.py	Complete derivation within the stated identifications. Knot spectral flow computed
Are EW quantities secretly fitted?	Import audit: top contribution from the framework's own G_F, m_t. Hadronic VP classified as measurement data, higher-order pieces as loop mathematics. No free theory parameters enter	ew_internal.py	Audited
Is the algebraic wiring arbitrary?	(7,26) bridge is the unique nontrivial boson of G₂(1) × F₄(1). Face channels confined. Sector graph and cycle inventory derived from the modular tensor category	wiring_theorem_probe.py	Derived
Can comparison data leak into predictions?	test_blindness.py perturbs reference values and requires predictions to remain identical. AST audit of reference.py access patterns	test_blindness.py	Enforced
Does the knot form in 3D?	Relative-mode RMS grows ×15.6 under G₂ coupling, flat under g₁ = 0 control. Common-mode shadow forms at the core	skyrmion_3d.py	Derived
What does the classical substrate claim, exactly?	The layer scoping: the classical field supplies the slots (a stable loop, three bound modes, channel lifting). The quantized bookkeeping supplies the ratios. A classical measurement of Q = 2/3 falsifies the scoping	circulant_normalization.py	Committed with falsifier

Where the questions end

The engine takes one number, and even that number is a unit name rather than physics: the ratio m_e/M_Pl is itself derived, so the dimensionless output takes no input at all. How strongly the substrate interferes is not a second input either. It is the thing being derived, one channel at a time. What the axiom cannot reach is why the amplitude is nonzero at all. Self-presence is universal, not conditional: the empty world is present to itself too, and precisely because it is nothing the self-touching generates no amplitude. That is not an exception to the axiom but its trivial fixed point, the zero orbit of x = b + W(x), zero in and zero out. Nothing does not escape the axiom. It just does not close down the road.

Contributing

Verification, critique, tooling, and candidate derivations meeting the promotion bar (a committed edge menu and stated kill conditions) are welcome. Corrections with anything adjustable in them are not, because there is nothing here to tune.

Papers

• The Innocent Lepton (10.5281/zenodo.19899091)
• One Substrate, Three Generations (10.5281/zenodo.20069456)
• The Echo of Standing Waves (10.5281/zenodo.20144381)
• The Octavian Coherence Gate: The Four Irreducible Integers of the E₈⁽¹⁾ ⊃ G₂⁽¹⁾ × F₄⁽¹⁾ Conformal Embedding (10.5281/zenodo.20493955)

Cite as: Kahsay, Kibrom Kidane (2026). One-field. https://github.com/kuwrom/one-field (the repository supersedes the papers as the living canon). See also CITATION.cff.

License

MIT. See LICENSE.